When running a sprint, the racers may be aided or slowed by the wind. The wind assistance is a measure of the wind speed that is helping push the runners down the track. It is much easier to run a very fast race if the wind is blowing hard in the direction of the race. So that world records aren't dependent on the weather conditions, times are only recorded as record times if the wind aiding the runners is less than or equal to 2 meters per second. Wind speed for a race is recorded by a wind gauge that is set up close to the track. It is important to note, however, that weather is not always as cooperative as we might like. The wind does not always blow exactly in the direction of the track, so the gauge must account for the angle the wind makes with the track. Suppose wind speed and direction is given by vector p = <8,5> and the track is in direction of vector q = <12,-4>. Determine if any race under these conditions could qualify for the record books.

Question

When running a sprint, the racers may be aided or slowed by the wind. The wind assistance is a measure of the wind speed that is helping push the runners down the track. It is much easier to run a very fast race if the wind is blowing hard in the direction of the race. So that world records aren't dependent on the weather conditions, times are only recorded as record times if the wind aiding the runners is less than or equal to 2 meters per second. Wind speed for a race is recorded by a wind gauge that is set up close to the track. It is important to note, however, that weather is not always as cooperative as we might like. The wind does not always blow exactly in the direction of the track, so the gauge must account for the angle the wind makes with the track. Suppose wind speed and direction is given by vector p = <8,5> and the track is in direction of vector q = <12,-4>. Determine if any race under these conditions could qualify for the record books.

Answer 1

p in quadrant 1, tan theta above x axis = 5/8 so angle = 32 deg

q in quadrant 4, tan angle below x axis = 4/12 = .333 so angle = 18.4 deg
total angle between wind and track = 32+18.4 = 50.4 deg
so
wind component in direction of track = |p| cos 50.4
|p| =sqrt (64 + 25) = 9.43 m/s
9.43 cos 50.4 = 6 m/s which is way more than 2 m/s

Answer 2

You are welcome.

Answer 3

Thank you this makes sense!

Answer 4

To determine if any race under these conditions could qualify for the record books, we need to calculate the wind assistance, taking into account the wind speed and direction.

First, let's normalize vectors p and q to calculate their respective magnitudes. The magnitude of a vector is the length of the vector and can be computed using the formula:

||v|| = sqrt(v1^2 + v2^2)

For vector p = <8, 5>:
||p|| = sqrt(8^2 + 5^2) = sqrt(64 + 25) = sqrt(89)

For vector q = <12, -4>:
||q|| = sqrt(12^2 + (-4)^2) = sqrt(144 + 16) = sqrt(160)

Next, we need to calculate the angle between vectors p and q. This can be done by using the dot product formula:

p · q = ||p|| * ||q|| * cos(theta)

where theta is the angle between p and q.

Since p and q are orthogonal (perpendicular), their dot product will be zero:

0 = sqrt(89) * sqrt(160) * cos(theta)

We can rearrange the equation to solve for cos(theta):

cos(theta) = 0 / (sqrt(89) * sqrt(160))

Since the cosine of any angle is always between -1 and 1, the wind cannot blow exactly in the direction of the track. This means the wind assistance cannot exceed 2 meters per second for the race to qualify for the record books.

Therefore, based on the given wind speed and track direction, it is not possible for any race under these conditions to qualify for the record books.